Cut a solid along an edge
Try using the Bisect tool.

Click Numpad 5 to enter into Orthographic View, then click Numpad 7 to go to top view.

Click A to select everything, then click Spacebar and type Bisect.

Click and drag across the cylinder while it is selected. This will cut a line into it along where you clicked.

Next, on the left tab, under bisect, make sure Fill and Clear Inner (or Clear Outer) are checked, or uncheck them if you simply want to add the edges along the mesh.